Engineering the temporal dynamics of all-optical switching with fast and slow materials

All-optical switches control the amplitude, phase, and polarization of light using optical control pulses. They can operate at ultrafast timescales – essential for technology-driven applications like optical computing, and fundamental studies like time-reflection. Conventional all-optical switches have a fixed switching time, but this work demonstrates that the response-time can be controlled by selectively controlling the light-matter-interaction in so-called fast and slow materials. The bi-material switch has a nanosecond response when the probe interacts strongly with titanium nitride near its epsilon-near-zero (ENZ) wavelength. The response-time speeds up over two orders of magnitude with increasing probe-wavelength, as light’s interaction with the faster Aluminum-doped zinc oxide (AZO) increases, eventually reaching the picosecond-scale near AZO’s ENZ-regime. This scheme provides several additional degrees of freedom for switching time control, such as probe-polarization and incident angle, and the pump-wavelength. This approach could lead to new functionalities within key applications in multiband transmission, optical computing, and nonlinear optics.


S3. Steady-state optical properties of titanium nitride and aluminum doped zinc oxide
The TiN films are measured with the variable angle spectroscopic ellipsometry (VASE) at angles of 50 and 70 o .We used a Drude Lorentz model to describe the optical properties of the TiN, with one Drude oscillator and two Lorentz oscillators.
where ε1 and ε2 are the real and imaginary parts of the dielectric permittivity, ħ is the reduced Plank constant,  is the probe angular frequency, and  ∞ is an additional offset.The negative (Drude) part shows the contribution of the free carriers to the permittivity:  0 is the square of the plasma frequency (in eV) and  0 is the damping factor of the Drude oscillator.The effect of bound electrons and interband transitions is modeled by a summation of the Lorentz oscillators, with  k ,  k , and   being the amplitude, broadening, and center energy of individual Lorentz oscillators.

S4. Temporal dynamics of TiN
Fig. S1 shows the temporal dynamics of TiN when pumped with 325 nm light at a pump fluence of 1.5 mJ/cm 2 .Prior work by George et al. and Diroll et al 1,2 show TiN to have an ultrafast electron-phonon relaxation rate that is see when probed near the ENZ point.Our experiment shows similar results, as seen from the sharp initial decay of the probe at 505 nm followed by the slower decay (black curve).Far from the ENZ, the ultrafast component is no longer seen, and the slower components span a nanosecond timescale (green graph).For this study, we focused on the slower dynamics of the TiN films.The overall dynamics is fitted with a two time-constant model, comprising of two exponential components, with time constants of the faster component of ≈20 ps for TiN followed by a slower time constant of ≈250 ps, both attributed to lattice cooling via diffusion and the radiation of heat from the measured sample spot, including through available interfaces.

S5. Temporal dynamics of AZO on Si with different powers, and that of AZO on fused silica
We perform pump-probe spectroscopy on the AZO films at various pump fluences.Fig. S2a shows the normalized reflectance spectra of AZO at various pump fluences.The overall decay time is invariant with increased pump fluence.Prior studies attribute this ultrafast response time to defect-assisted Shockley-Read-Hall recombination, a common recombination pathway in heavily doped TCOs [3][4][5] .
The carrier relaxation time of doped oxides may also be dependent on the growth substrate 6 .To study how much the recombination time depends on the substrate, we grew AZO fused silica.Pump probe spectroscopy of the AZO grown on fused silica shows it to have a recombination time under 10 ps.
where , ,  are the relaxation fitting constants summarized in Table S3.

S6. Metasurface response as a weighted sum of the material responses
Having extracted the temporal responses of the individual films, the response of the metasurface at a specific wavelength can be modeled as a weighted sum of the temporal response of the TiN and the AZO films.S3 shows the fitted graphs of the experimental data at each wavelength.For these fits, we used the time constants of TiN at 505 nm and of AZO at 1200 nm.Incorporating the wavelength dependent variations of the time constants can be expected to give better fits.Fig. S3b shows the relative weight of the contribution of TiN to the overall response α/(α + ) (red curve), overlapped with the simulated absorbance of the TiN (AbsTiN) relative to the total absorbance (AbsTiN+AZO) at each wavelength.The trends show a strong correlation.The absorbance of probe in a medium is strongly affected by the light-matter interaction between the probe and the medium.
This shows that the time constants of the recombination are indeed influenced strongly by the light-matter interaction of the probe with the individual layer.S7.Controlling the device switching speed by changing the probe angle of incidence, polarization, and pump wavelength.
In our system, for a fixed wavelength, changing the probe angle of incidence decreases the proportion of light interacting with the fast (AZO) material and increases the interaction with the slow (TiN) material, thus slowing the effective switching rate down (Fig. S4a).In Fig. S4a, for the 30 o angle of incidence, more of the electromagnetic field interacts with TiN, the dynamics thus follow the slower TiN dynamics, which take several nanoseconds to eventually fall to zero.For the 30 o case, the nonzero term takes several nanoseconds to fall to zero because of the slow TiN response 1 .
Similarly, we also observe a nanosecond tail in the transient response with an s-polarized probe at the NIR wavelength range (Fig. S4b).The sign reversal and the increase after 1 ps result from the shift and the broadening of additional Fabry-Perot resonances due to the injection of photoexcited hot-carriers, which interact with the reflectance signal.The complex interplay of resonance shift and broadening that result in the tens of picosecond to hundred picosecond components likely arises from hot carrier injection as well as thermal expansions that alter resonant conditions.In this case, s-polarization presents higher sensitivity to these changes.A more rigorous investigation of similar dynamics in ENZ films is reported in the work by Fruhling et al 7 .
Figure S4.(a) At an angle of 30 degrees, a greater proportion of the probe interacts with the TiN, resulting in a nanosecond tail (b) A similar, slower response can also be seen by moving from a p-polarized to an s-polarized probe at the same wavelength.There is also a sign reversal of the signal change that can be attributed to complex, non-equilibrium dynamics of photoinjected carriers that result in the shift and broadening of additional resonances in the structure.
At the visible wavelengths, where most of the light will primarily interact with the TiN layer, a fast response can still be achieved by utilizing a different pump which interacts more strongly with the faster (AZO) material, thus leading to a sub-picosecond response.The following experiment shows the device response with a visible probe at 488 nm wavelength (at 50 o angle of incidence) when pumped with an intraband pump of 1400 nm wavelength (p-polarized, 70 o angle).At this configuration, the AZO absorbs most of the signal, resulting in a fast, femtosecond response (Fig. S5, solid black line).Note that the TiN response time at the same probe wavelength is in nanoseconds, regardless of the pump wavelength (dashed lines).
Figure S5.An intraband pump at 1400 nm and a 70º angle of incidence results in a femtosecond device response to a visible probe at 488 nm.This is because the pump is mostly absorbed in the AZO layer, resulting in a femtosecond device response, even though most of the probe interacts with the TiN layer.The dashed lines show the responses of the TiN film (red) with a 1400 nm pump, showing the slower dynamics, and the device dynamics with a 325 nm pump."

Figure
Figure S2.(a) Transient reflectance spectra of AZO on Si at different pump-fluences at 1200 nm (b) Transient reflectance spectrum of AZO on Si overlapped on that of AZO grown on fused silica, showing similar response times.

Figure
Figure S3 (a) Experimentally obtained temporal response of the metasurface (normalized) at different wavelengths (dots) overlapped with the fits (solid lines).The maximum pump-probe overlap for each experiment was alligned to center at 1 ps, and each curve is vertically translated by 0.3 from the next for clarity.(b) Relative weight of the TiN contribution to the sum of TiN and AZO contributions (red), and the relative absorbance of TiN as a fraction of the total absorbance of the metasurface (black)

Table S1 .
Drude-Lorentz parameters for the as-deposited TiN and AZO films

Table S2 .
Fitting parameters of TiN transient reflectance modulation Figure S1.Normalized transient reflectance of TiN at different wavelengths

Table S3
Fitting parameters of AZO transient reflectance modulation at a wavelength of 1.2 μm weight factor of TiN (with time constants,  1 = 17.3 ,  2 = 287  and constants  = 0.33,  = 0.26, from TableS.2 at 505 nm) and  is a weight factor of the AZO (with time constant  = 0.72  and constant  = 1.98 from TableS.3 at a fluence of 0.18mJ/cm 2 ). comprises much slower lattice cooling effects, together with detector noise arising from probe fluctuations.Figure